TOWARDS A CLASSIFICATION OF MULTI-FACED INDEPENDENCES: A COMBINATORIAL APPROACH

. We determine a set of necessary conditions on a partition-indexed family of complex numbers to be the “highest coefficients” of a positive and symmetric multi-faced universal product; i.e. the product associated with a multi-faced version of noncommutative stochastic independence, such as bifreeness. The highest coefficients of a universal product are the weights of the moment-cumulant relation for its associated independence. We show that these conditions are almost sufficient, in the sense that whenever the conditions are satisfied, one can associate a (automatically unique) symmetric universal product with the prescribed highest coefficients. Furthermore, we give a quite explicit description of such families of coefficients, thereby producing a list of candidates that must contain all positive symmetric universal products. We discover in this way four (three up to trivial face-swapping) previously unknown moment-cumulant relations that give rise to symmetric universal products; to decide whether they are positive, and thus give rise to independences which can be used in an operator algebraic framework, remains an open problem.


Introduction
At the latest with Voiculescu's invention of freeness [Voi85], it became apparent that the "obvious" extension of classical stochastic independence, tensor independence, is not the only and not always the most suitable concept in inherently noncommutative situations.In fact, Boolean independence (not yet under this name) has already featured much earlier in the work of von Waldenfels [vW73, vW75].Those "noncommutative independences" share many properties with classical stochastic independence and tensor independence.In particular, under the assumption of independence, mixed moments are uniquely determined and can be calculated from marginal moments (also giving rise to an associated convolution product for probability measures on the real line).Another interesting independence is monotone independence, which was discovered by Muraki [Mur01]; this is a non-symmetric independence relation.
An extremely useful tool when dealing with random variables which have all moments are the corresponding cumulants.The theory of free cumulants, linearizing free additive convolution, was developed by Speicher [Spe94], see also the book by Nica and Speicher [NS06]. 1 Boolean cumulants were formalized by Speicher and Woroudi [SW97].Understanding the monotone cumulants took a bit longer, many questions were answered by Hasebe and Saigo [HS11].The problem in the monotone case is that independence is not in general characterized by vanishing of mixed cumulants.This is directly related to the non-symmetric nature, as becomes apparent when interpreting moment-cumulant relations via exponential and logarithm maps, as is done in related but different settings by Manzel and Schürmann [MS17] (Hopf algebraic) or Ebrahimi-Fard and Patras [EFP15] (shuffle-algebraic); non-zero mixed cumulants can appear in the Campbell-Baker-Hausdorff series.
Since the work of Speicher [Spe97], Ben Ghorbal and Schürmann [BGS02], and Muraki [Mur02,Mur03], we know that the five independence relations for noncommutative random variables, tensor, free, Boolean, monotone and antimonotone independence, are indeed very special.For these independences, the joint distribution of independent random variables is obtained from the marginal distributions by means of a "universal product", i.e. a product operation which fulfills a number of natural conditions, including associativity and universality (i.e. in a specific sense not dependent on the concrete realization of the noncommutative random variables) and a "factorization for length 2"-condition; and they are the only ones with this property. 2Replacing that "factorization for length 2"-condition by a positivity condition, a decade later, Muraki [Mur13] proved a similar result with a much simpler proof, while at the same time using a much better motivated assumption, namely that the product operation restricts to a product operation for states on augmented * -algebras. 3This kind of positivity is also the right condition to study quantum Lévy processes on dual groups in the sense of Ben Ghorbal and Schürmann [BGS05], see also [SV14], where Schoenberg correspondence between convolution semigroups of states and conditionally positive generators is proved in this context.In 2014, Voiculescu [Voi14] introduced a new nontrivial extension of free independence, bifreeness, for sequences of pairs of random variables, or pairs of faces as Voiculescu called the general underlying framework.Taking up on this idea, more examples of 2-faced or, more generally, multi-faced independences have been discovered [Liu19,Liu18,GS19,GHS20,Ger23].The general theory of multi-faced universal products from which those independences can be obtained was established by Manzel and Schürmann [MS17].It turned out that not all of the examples fulfill the natural positivity condition.Positivity is still enough to assure Schoenberg correspondence in this generalized setting, see [Ger21].In an effort to classify positive multi-faced universal products, two routes have been taken.In [GHU23], Gerhold, Hasebe, Ulrich completely classified 2-faced universal products which have a natural representation on the tensor product or the free product Hilbert space of the GNS spaces of the factors.In Varšo's PhD thesis [Var21], he proved that there are at most 12 two-faced universal products which fulfill additional assumptions of symmetry and a "combinatorial" moment cumulant relation (i.e.determined by a subset of all two-faced partitions, where more generally weights on two-faced partitions can appear). 4In this article we present, simplify, and extend those results of [Var21].
A single-faced independence can trivially be regarded as a two-faced independence, and every two-faced independence is a certain kind of mixture of two single-faced independences.However, neither do those two single-faced independences determine the two-faced independence, nor is it obvious that any combination of single-faced independences can be combined in any way to form a two-faced independence. 5The main result of this article is to present a family of twofaced symmetric universal products such that every positive symmetric two-faced universal product must belong to that family, we call them candidates.This is achieved in three steps.First, we prove necessary conditions for a family of weights on ordered partitions to be the highest coefficients of a positive multi-faced universal product (Theorem 5.3); second, we determine all permutation invariant weights (= weights on non-ordered partitions) which fulfill those properties (Corollary 6.11), we call such weights here admissible; third, we prove that admissible weights are always the highest coefficients of a (uniquely determined) symmetric multi-faced universal product (Theorem 8.2).The family of candidates consists of (identifying an independence with its underlying universal product, and disregarding the difference between a 2-faced independence and its image under swapping the faces) • 2-faced continuous 1-parameter deformations of free, tensor and bifree independence (positivity is proved in [GHU23]), • a tensor-free independence (positivity is not known), • a new free-free and a new tensor-tensor independence, different from the trivial ones, bifreeness, and their deformations (positivity is not known), • tensor-Boolean, free-Boolean and Boolean independence; positivity for those is also covered in [GHU23], for free-Boolean it was first shown by Liu [Liu19] and for Boolean independence positivity is of course well-known.We call the independences which are not realized in [GHU23], i.e. those whose positivity is yet unknown, exceptional.
We prove many of the preliminary results for the general symmetric multi-faced case.Theorem 5.3, where we find necessary conditions on weights to arise as highest coefficients of a universal product, is even formulated for not necessarily symmetric products and could be used as a starting point for a more general classification including multi-faced universal products based on monotone independence, such as for example bimonotone independence (of type II) as defined in [Ger23,GHS20].
It easily follows from the main result that there are no non-trivial positive and symmetric trace preserving universal products (Remark 6.12) and that tensor independence and bifreeness are the only two positive symmetric 2-faced independences which allow to define a convolution of probability measures on R 2 (Remark 6.13).
Among our additional results, we characterize when a positive symmetric multi-faced universal product is unit preserving (Theorem 9.7), i.e. when it can be defined consistently for arbitrary unital algebras (in the other cases, the product operation is only defined for linear functionals on augmented algebras).This is indeed the case for the three continuous families and the four (three up to swapping the faces) exceptional cases.Furthermore, we establish a simplified mixed moment formula for the special combinatorial case where the highest coefficients are only 0 or 1, so that the moment cumulant relation is simply governed by a specific set of partitions (Theorem 8.4).
The outline of the article is as follows.In Sections 2 to 4, we introduce the basic concepts, in particular multi-faced universal products and multi-faced partitions.In Section 5 we prove the necessary conditions for a family of weights to be the highest coefficients of a positive multi-faced universal product (symmetric or not).In Section 6 we show that those necessary conditions allow us to obtain a concrete list of candidates for symmetric and positive two-faced universal products.In Section 7 we give an introduction to Manzel and Schürmann's cumulant theory, adapted to the relevant special case of symmetric multi-faced independences.In Section 8 we prove, using cumulants, that in the symmetric case the conditions exhibited in Section 5 are sufficient to reconstruct a universal product in the algebraic sense (with a simplified formula in the combinatorial case), but it remains open whether these universal products are automatically positive.Finally, we characterize in Section 9 which universal products in our list are unit preserving.In Section 10 we name four tasks which have to be completed in order to achieve a complete classification of positive multi-faced universal products.
A comparison between this article and corresponding results in Varšo's PhD thesis [Var21] can be found in Appendix A.

Preliminaries and notation
We will have to deal a lot with tuples of all kinds, so we introduce some useful notation.Let X and Y be arbitrary sets.For any natural number n, denote by [n] the set {1, . . ., n}.For an n-tuple t = t(1), . . ., t(n) ∈ X n and a subset I = {i 1 < . . .< i k } ⊂ [n], we define the restricted tuple t ↾ I := t(i 1 ), . . ., t(i k ) .Two tuples t ∈ X n , s ∈ Y n of the same length may be combined to form the tuple t × s ∈ (X × Y ) n with (t × s)(i) = t(i), s(i) , and conversely, every tuple in (X × Y ) n is of that form.The set of n-tuples of arbitrary length n is denoted X * = n∈N0 X n .When a set X does not carry any multiplicative structure, we might use the word notation, t(1) • • • t(n) := t(1), . . ., t(n) ∈ X n .The entries of a tuple t might be written t i instead of t(i) from time to time; or we might use t as a shorthand for (t 1 , . . ., t n ) without further comment when the t i have been around before.
An algebra means a complex associative algebra, not necessarily unital.The free product of algebras A 1 , A 2 is denoted A 1 ⊔ A 2 , reminding of the fact that this is the coproduct in the category of algebras: for arbitrary algebra homomorphisms h i : it should always be clear from the context which codomain is meant.
For a vector space V , we denote by T 0 (V ) = n∈N V ⊗n the (non-unital) free algebra over V .We will identify T ) by extending them as the 0-functional to the canonical complement, i.e.
In particular, this convention applies to the direct sum of two linear functionals ψ i : T 0 (V i ) → C, i.e. we identify ψ 1 ⊕ ψ 2 with the linear functional on T 0 (V 1 ⊕ V 2 ) given by (1) The unital free algebra is denoted T (V ) = n∈N0 V ⊗n , and this unital algebra is the unitization of T 0 (V ).
For the rest of this article, if not explicitly mentioned otherwise, F denotes a fixed finite set, whose elements we call faces or colors.We could of course assume F = [m] for m ∈ N, but since there will be a lot of integers around, we prefer to use more abstract symbols.We mostly use squared symbols such as , to denote arbitrary elements of F. If there are exactly two faces, we assume F = { , }.
A multi-faced (or F-faced6 ) algebra is an algebra A that is freely generated by given subalgebras A , ∈ F (the faces of A), i.e. the canonical algebra homomorphism ∈F A → A is an isomorphism; this is indicated by writing A = ∈F A .A multi-faced algebra homomorphism is an algebra homomorphism h : A → B between multi-faced algebras A, B with h(A ) ⊂ B .We consider the free product of multi-faced algebras again a multi-faced algebra with faces (A ⊔ B) := A ⊔ B .Note that the free product of multi-faced algebras is the coproduct in the category Alg F of multi-faced algebras with multi-faced algebra homomorphisms, i.e. for every pair of multifaced algebra homomorphisms h i : A i → B there is a unique multi-faced algebra homomorphism h 1 ⊔ h 2 : A 1 ⊔ A 2 → B restricting to h i on A i , respectively for i = 1, 2.
A multi-faced * -algebra is a multi-faced algebra with an involution such that each face is a * -subalgebra.Of course, the free product of multi-faced * -algebras is again a multi-faced * -algebra in the obvious way and the free product of multi-faced * -homomorphisms is a * -homomorphism.
We say that a linear functional φ : A → C defined on a multi-faced * -algebra is a restricted state if its unital extension to the unitization of A is a state (or, equivalently, positive).

Universal products
Definition 3.1 (Cf.[Ger21,Rem. 3.4]).A multi-faced universal product is a binary product operation for linear functionals on multi-faced algebras (with an a priori fixed set of faces F) which associates with functionals φ 1 , φ 2 on multi-faced algebras A 1 , A 2 , respectively, a functional • positive if the product of restricted states on multi-faced * -algebras is a restricted state on the free product * -algebra.
Note that we made several implicit identifications between isomorphic free products in the last definition.For a more detailed discussion see [Ger21].
Universal products have been invented to encode independences.In the single-faced case, this has been worked out by Ben Ghorbal and Schürmann [BGS02].The multi-faced case is covered by [MS17] together with the categorical considerations from [Fra06] and [GLS22].In a nutshell, given a universal product ⊙ and a linear functional Φ on an algebra A, algebra homomorphisms or, in other words, if the joint distribution of the noncommutative random variables j κ coincides with the universal product of their marginal distributions.This induces the usual definitions of independence for F-tuples of elements or of subalgebras of A. In the remainder of this article, we will not work with the independences themselves, but solely with the underlying universal products, so we refrain from giving more details here.
We will make extensive use of the "Central Structural Theorem" for universal products [MS17, Theorem 4.2].Before we present a simplified version of it adapted to the special case of positive multi-faced universal products, we introduce some more notation and give an example.
Let A 1 , . . ., A k be multi-faced algebras and we identify the A i with subalgebras of their free product).For Note that the A s are not necessarily pairwise disjoint.7 Elements of [k] n are referred to as block structures and elements of F n are called face structures. For We call a set partition π of [n] adapted to s, and write π ≺ s, if the following two conditions are met: • each block β ∈ π is contained in some β κ (s); in other words, π is a refinement of the set partition σ = {β 1 (s), . . ., β k (s)} (to adhere strictly to the usual definition of set partition, empty blocks should be removed from σ) • if s(i) = s(i + 1), then i, i + 1 belong to the same block of π.Note that, obviously, σ is the maximal partition (w.r.t.refinement order) adapted to s.
Given a multi-faced universal product ⊙, we define its linearized part as (that this expression is well-defined should be understood as part of the following theorem).as follows, abbreviating ⟨b⟩ := φ κ (b) for b ∈ A κ : Consequently, the linearized part is given by Note how the summands in the full expansion in Example 3.2 correspond to partitions adapted to s; the product element a is divided into some sort of "subproducts" which are then evaluated in the appropriate φ κ .This general pattern is made precise in the following theorem and allows to describe a universal product in terms of the complex coefficients appearing in each summand, which are independent of the involved linear functionals, algebras and algebra elements.Let ⊙ be a positive multi-faced universal product and k ∈ N. Then there are unique coefficients α π s , s ∈ ([k] × F) * , π ≺ s, such that , for all linear functionals φ κ : ) and all a ∈ A s , indicates that the product is to be taken in the same order as the factors a j appear in the product a = a 1 • • • a n .) Putting α s := α σ s (σ the maximal partition adapted to s), the linearized part is given by The α π s are called coefficients of ⊙ and the α s are called highest coefficients of ⊙.Proof.First assume that s ∈ ([k] × F) n is alternating, i.e. s(i) ̸ = s(i + 1) for i = 1, . . ., n − 1.By [MS17, Rem.4.3], the formula given in [MS17, Th. 4.2] can be applied.For a positive universal product, [MS17, Rem.4.4] implies that there is only one summand for each π ≺ s, corresponding to the "right-ordered coefficient" (i.e. the a j are multiplied in the same order in which they appear as factors in a) associated with π and s, denoted α π s in this article.If s is not alternating, then we define α π s := α π s , where s is the alternating tuple obtained from s merging repeating entries into one, and π the set partition adapted to s induced by π in the obvious way.By universality it follows that (2) extends to all s for i = 1, . . ., m (r 0 := 0), one finds that, using universality for the first equality, For each ρ ≺ s, one can easily construct multi-faced algebras and linear functionals φ κ : A κ → C and an element a ∈ A s in such a way that and, thus, . This shows uniqueness of the coefficients.Equation (3) follows from Equation (2) because the summand corresponding to the maximal partition σ is the only one which is linear in each φ κ .□ Obviously, the family of coefficients determines the universal product.In fact, it follows from the cumulant theory developed in [MS17] that the highest coefficients alone are already enough to determine the universal product.We will come back to this in Section 7.
To end this section, we show that the highest coefficients can be recovered from the linearized part of a universal product using only linear functionals of a particularly well-behaved kind.
Definition 3.4.A restricted state φ : A → C on a multi-faced algebra A is called trivially multifaced if for all , ∈ F there exists a * -isomorphism a → a : A → A with φ(ab c) = φ(ab c) for all b ∈ A and all a, c in the unitization of A.
Lemma 3.5.For every s = b×f ∈ ([k]×F) * , there are trivially multi-faced restricted states φ κ on multi-faced * -algebras Proof.Define A κ := C and A κ := ∈F A κ .Then φ κ = ∈F id : A κ → C is a state, in particular a restricted state, and trivially multi-faced.Put a κ := 1 for all κ ∈ [k] and all ∈ F. Now it is easy to see that φ κ (a the first claim is obvious and the second claim follows from Theorem 3.3.□

Partitions
In general, a multi-faced set is a set S together with a map f : S → F, the face structure of S. The subsets S := f −1 ({ }) are called the faces of S. A multi-faced subset of S is just a subset of the underlying set viewed as a multi-faced set with respect to the restricted face structure.
In this article, we only deal with multi-faced sets whose underlying set S is finite and totally ordered; these properties are implicitly assumed whenever we write about multi-faced sets in the following.
Any word , (which we identify with the word f ) thus turning [n] into a multi-faced set, denoted by [n] f .Conversely, we associate with a multi-faced set S = ({s We choose this on first sight odd notation because the word f plays the same role as the number of elements of a set plays in the single-faced case in the moment-cumulant formulas we are aiming at.
Let S be a multi-faced set and ∼ an equivalence relation such that • the equivalence classes are intervals, • f is constant on equivalence classes.
Then we understand the quotient S/∼ as a multi-faced set with the induced total order and face map.
Example 4.1.We briefly discuss the two situations that will appear several times in this article.
(1) Let f ∈ F n be a face and ∼ the equivalence relation on [n] that identifies two neighboring points i, i + 1 in the same face, i.e. f (i) = f (i + 1).In this case we write f /(i ∼ i + 1) for the quotient [n] f / ∼ and denote its elements ℓ instead of {ℓ} for the trivial equivalence classes of ℓ ∈ [n] \ {i, i + 1} and {i, i + 1} for the two-element equivalence class of i and i + 1.
(2) Let S be a multi-faced set and ∼ the equivalence relation whose equivalence classes are the maximal intervals on which f is constant.We then call the quotient S red := S/∼ the reduction of S. In the reduction, neighboring points will always have different faces, so that no further quotienting is possible.
A partition of a multi-faced set S is a collection of multi-faced subsets whose underlying sets form a set partition.The set of all partitions of a multi-faced set S is denoted P(S).An ordered partition of S is a partition of S together with a total order between the blocks.The set of all ordered partitions is denoted P < (S).
For a word f ∈ F n , we put P(f ) := P([n] f ) and P < (f ) := P < ([n] f ).We also denote This can be nicely drawn as an arc diagram, π = .
In the following we will not distinguish between a partition and its arc diagram.In this article, we mostly use arc-diagrams to denote partitions in P, i.e. without a block-order; the height of the blocks is then completely arbitrary.For a partition in P < , the height of the block corresponds to the order between blocks.If the underlying set S is not of the form [n] f (typically because it was obtained as a quotient), we draw the diagram for the corresponding partition of |S|.P(f ) is a partially ordered set by the order of reverse refinement.The maximum and minimum of P(f ) are denoted 1 f and 0 f , respectively, i.e. 1 f is the one-block partition and in 0 f all blocks are singletons.
There is a canonical bijection between P(S/∼) and the set of π ∈ P(S) such that equivalent points of S lie in the same block of π.
For a multi-faced partition π, consider the equivalence relation ∼ defined on the underlying multi-faced set S by s ∼ t :⇐⇒ all r ∈ S with s ≤ r ≤ t have the same color and belong to the same block of π.In other words, ∼ is the equivalence relation whose equivalence classes are the maximal intervals I of S which fulfill the following two properties: • f is constant on I; • all elements of I belong to the same block of π.We define the reduction of π as the induced multi-faced partition π red on S/∼.For example, red = .
Then π red will not have neighboring legs that are in the same face and in the same block.For π ∈ P < , the block order remains unchanged.For a multi-faced set S, we define its mirror image S as the set with one element s for each s ∈ S (so that s → s is a bijection) with the face structure f (s) := f (s) and reversed order, i.e. s ≤ t ⇐⇒ s ≥ t.For π ∈ P(S), we put π ∈ P(S) as the set partition with a block β = {s 1 , . . ., s n } for each block β = {s 1 , . . ., s n } ∈ π.For example, we use the convention that k := n − k + 1 (i.e.we identify k with its image under under the unique strictly increasing map the mirror image of f .This is clearly in accordance the diagrammatic representation.If π = {β 1 < . . .< β k } ∈ P < , then π is defined as before together with the (non-reversed!) block order β 1 < . . .< β k .Finally, we introduce a notation for uniting blocks.Let π = {β 1 < . . .< β k } ∈ P < (S) with blocks β i , β i+1 that are nearest neighbors for the order on π.Then we define π βi⌣βi+1 := {β 1 < . . .< β i−1 < β i ∪ β i+1 < . . .< β k }.Similarly, for π ∈ P(f ) and arbitrary blocks , be face structures and f their concatenation, i.e. f (m Given partitions π i ∈ P(f i ), we define their concatenation as the partition π ∈ P(f ) which has for every block β ∈ π i with i ∈ [n] a block β := {ℓ : Roughly speaking, π restricts to π i on the legs corresponding to f i .For example, the concatenation of π 1 = and π 2 = is π = .We do not define here the concatenation of ordered partitions.
For a family of numbers (as it is for example obtained from a universal product by Theorem 3.3) and π = {β 1 < . . .< β k } ∈ P < (f ) an ordered multi-faced partition with k blocks, we define In this way, we associate with each universal product a family of weights on ordered partitions, and we say that the weights of a universal product are its highest coefficients.Note that such weights are always monic.
We say that weights on ordered partitions α are invariant under permutation of blocks if In this case, define α π for a non-ordered partition π = {β 1 , . . ., β k } ∈ P(f ) simply as the value α {β1<...<β k } for an arbitrary ordered partition with the same blocks as π.In this way, we can identify weights on partitions and weights on ordered partitions which are invariant under block permutation.
Remark 5.2.It is easy to check that the weights α coming from a universal product according to Theorem 3.3 are invariant under permutation of blocks if and only if the universal product is symmetric.
The question we wish to answer is the following: under which conditions on the weights α is there a (positive) universal product ⊙ with highest coefficients α?The next theorem yields some necessary conditions.
Theorem 5.3.Let ⊙ be a positive multi-faced universal product.Then the highest coefficients fulfill: (iv) Suppose π ∈ P < (f ) has blocks β 1 < β 2 that are nearest neighbors for the order of π and have neighboring legs in the same face, i.e. there exist (v) α π = α σ whenever π and σ only differ in the faces of extremal legs.
the beginning of this section.By Lemma 3.5, we can express each coefficient α π as with a ∈ A π := A sπ and (φ We will freely use this notation in the rest of the proof. (i) follows from the restriction property in Definition 3.1.(iii) holds by definition of the nonreduced coefficients in the proof of Theorem 3.3.For (iv) we have to carefully analyse the linearized universal product.If π has neighboring blocks β 1 < β 2 with neighboring legs in face ∈ F, then Evaluating the full coefficient formula, Equation (2), for the universal product of the k − 1 functionals because the two factors a r , a r+1 are from the same block and face and therefore have to be treated as one.Summands with more factors containing ψ i vanish in the linearization procedure.Therefore, we obtain So far, we have not made significant use of positivity (except that we assumed that wrong ordered coefficients vanish), but positivity is important to prove the remaining two properties.
(vi) follows easily from the fact that positive functionals are hermitian and a ∈ A π if and only if a * ∈ A π .All we have to do is choose some restricted states φ 1 , . . ., φ k and a ∈ A π with To show (v), assume that with a 1 ∈ A i and trivially multi-faced restricted states φ κ , this is always possible by Lemma 3.5.Then t κ φ κ is a restricted state for all t κ ≤ 1, and where a 1 is the image of a 1 under the isomorphism A i ∼ = A i making φ i trivially multi-faced.From this the statement for the first leg readily follows.For the corresponding statement for the last leg, we can either apply (vi) or perform an analogous computation.
For the rest of this article, we restrict ourselves to the symmetric case.As noted before, symmetry of the universal product is equivalent to invariance under block-permutation of its highest coefficients, and in this case we denote its highest coefficients α π with π ∈ P.
Observation 5.8.Let (α π ) π∈P be admissible weights.Then Π α = {π : α π ̸ = 0} is an admissible set of partitions.There are, however, admissible families with α π / ∈ {0, 1} for some π ∈ P. Indeed, Example 3.2 in particular shows that, for α the highest coefficients of the deformed tensor product Observation 5.9.A set Π ⊂ P of partitions is admissible if and only if Π contains the partitions (P-i) 1 f for all f ∈ F * (P-ii) for all ∈ F 2 and is closed under the following operations used in [Var21]: (P-iii) double a leg, including its color (P-iii)' merge two neighboring legs of the same color in the same block into one (P-iv) unite two blocks which have neighboring legs of the same color into one block, π → π β1⌣β2 (P-iv)' remember a two-block partition formed by two blocks with neighboring legs of the same color, π → {β 1 , β 2 } (P-iv)" replace a block of a partition from Π by a two-block partition from Π (of the same underlying multi-faced set as the original block) such that the blocks have neighboring legs of the same color, (π β1⌣β2 , {β 1 , β 2 }) → π (P-v) mirror a partition, π → π (P-vi) change color of an extremal leg of a partition from Π Given any partitions π 1 , . . ., π n ∈ P, we denote by ⟨π 1 , . . ., π n ⟩ the minimal admissible set of partitions that contains all π i .We say that ⟨π 1 , . . ., π n ⟩ is generated by π 1 , . . ., π n ; note that ⟨π 1 , . . ., π n ⟩ indeed consists of those partitions in P which can be obtained in finitely many steps by applying the operations of Observation 5.9 to the partitions 1 f (f ∈ F * ), ( ∈ F 2 ), and π 1 , . . ., π n .

Partial classification of symmetric positive independences
In this section we determine all admissible families (α π ) π∈P .Definition 6.1.Let π be a partition.
• A leg ℓ is called inner if there exist legs i < ℓ < j and a block β ∈ π with i, j ∈ β and ℓ / ∈ β.Otherwise it is called outer.• Two legs ℓ, ℓ ′ are called connected if they lie in the same block or if there is a sequence of blocks ℓ ∈ β 1 , . . ., β n ∋ ℓ ′ such that there is a crossing between β k and β k+1 .Roughly speaking, ℓ and ℓ ′ are connected if and only if one can move from ℓ to ℓ ′ going only along the lines of the diagram associated with π.
We start by describing some simple consequences of the defining properties of admissible families of coefficients.Lemma 6.2.Let (α π ) π∈P be admissible weights.
(3) α π = α σ when σ is obtained by replacing one leg ℓ by two copies and splitting the block β ∋ ℓ into β 1 and β 2 , where β 1 contains the first copy and all legs of β smaller than ℓ and β 2 contains the second copy and all legs of β larger than ℓ.We say that σ is obtained by splitting β at ℓ. (4) α π = α σ when σ is obtained replacing an arbitrary number of connected outer legs by a single outer leg of arbitrary color.We call this process collapsing the outer legs. Proof.
(1) This is easily proved by induction.For a two-block interval partition π, we can consecutively change color of the extremal legs and merge them with their neighboring legs until we reach α π = α = 1.It is worth noting that for this step we needed to change the color of both extremal legs.
(2) Clearly, it is enough to prove the claim for n = 2.We prove the claim by induction on the number of blocks |π|.If |π| = 2, then |π 1 | = |π 2 | = 1 and the three partitions are interval partitions, in particular We can assume without loss of generality that 2 ∈ β 2 belongs to a different block β 1 ̸ = β 2 ∈ π 1 and f (1) = f (2); if those conditions are not met, it does not change the coefficient to change the color of the first leg to match the color of the second leg and merge them into one until we are in the described situation.Now we find . Of course, |π β1⌣β2 | = |π| − 1, so we may assume that the statement holds for π β1⌣β2 which is the concatenation of π 1β 1⌣β2 and π 2 .Altogether, If |π 2 | > 1, we argue analogously, but we have to change the color of the last leg.
(3) We have α π = α σ α {β1,β2} , and α {β1,β2} = 1 by (1).(4) Decompose π into a concatenation of irreducible π 1 , . . ., π n , i.e. no π i can be deconcatenated any further.By Item (2), α π = α πi .Note that every outer leg of π is the outer leg of some π i and that connected outer legs are necessarily in the same block.For each π i , the outer legs can be collapsed by iteratively changing the face of the first or last leg to match the face of its successor or predecessor, respectively, and merging the legs using the fact that the weights don't change when we reduce the partition (Condition (iii) in Theorem 5.3).After collapsing the outer legs that way, the faces of the outer legs can be changed once more in such a way that the concatenation of the obtained partitions σ i is σ.It follows, using again Item (2), that α σ = α σi = α πi = α π .

□
It is worth noting that, in the proof of Item (2), we need invariance of the coefficients under changing the faces of both extremal legs.For example, the weights associated with bi-Boolean independence defined in [GS19] do not share this property.Lemma 6.3.Two admissible families coincide if and only if they coincide on 2-block partitions.
Proof.Assume that (α π ), (β π ) are admissible families with α σ = β σ for all 2-block partitions σ.By definition, the value on 1-block partitions is 1.Given an n-block partition π with n > 2, we alternatingly • change the color of the first leg to match the color of the second leg, cf.(v), • combine the first two legs into one if they belong to the same block, cf.(iii), to obtain a partition π such that the first two legs of π have the same color but belong to different blocks β 1 , β 2 .Then α π = α π , β π = β π by definition of admissible weights.Using (iv), we then have , where π β1⌣β2 is an (n − 1)-block partition and {β 1 , β 2 } is a 2-block partition.We can iterate the procedure until we obtain α π , β π as products of coefficients of the same sequence of 2-block partitions, thus proving the claim.□ Corollary 6.4.Two admissible families coincide if and only if they coincide on 2-block partitions of at most four legs.
Proof.Suppose that π = {β, γ} has more than 4 legs and that the third leg lies in β.Without loss of generality, we assume that the first leg and the second leg belong different blocks but the same face; if they would belong to the same block, they could be collapsed and the face of the first leg can simply be adapted to that of the second leg without changing the coefficient.Without loss of generality assume that 1 ∈ β.If all legs after the third leg belong to γ, they are necessarily outer and can be collapsed to reach a partition with four legs.If there is at least one leg from β after the third leg, then splitting β at the third leg yields a partition π = {β 1 , β 2 , γ} where β 1 = {1, 3} has two legs and β 2 = {3 ′ } ∪ (β \ {1, 3}) has exactly one leg less than β; here 3 ′ is the copy of 3 obtained from splitting such that 3 < 3 ′ .Now, α π = α π = α πβ 1 ⌣γ α {β1,γ} .Obviously, τ := {β 1 , γ} has strictly less legs than π.Since the first three legs of πβ1⌣γ belong to the same block, after collapsing those three legs, we get a partition σ with α σ = α πβ 1 ⌣γ which has one leg less than π (one leg more from the splitting are overcompensated by two legs less from collapsing).All in all, α π = α τ α σ , where both, τ and σ are two-block partitions with a strictly smaller number of legs than π.This procedure can be iterated until α π is expressed as a product of only 2-block partitions with at most 4 legs.□ Definition 6.5.We introduce shorthand notations for the basic coefficients, where , ∈ F: (Note that ξ = ξ , obviously, and ν = ν , because we can merge neighboring legs of the same face.)Corollary 6.6.Two admissible families coincide if and only if they have the same basic coefficients.
(1) This follows as in the single-faced case, see [Spe97].Alternatively, this follows easily as a special case = from the items below.(2) Consider .Split the inner -leg and merge it's copy with the outer block to obtain α = α α .The other cases work analogously.
(3) First note that α = α = |ν | 2 .This leads to (4) This follows from (5) Reusing parts of the calculation above, we find NC denotes the set of all noncrossing partitions, • binoncrossing if for all i < j < k < ℓ and blocks • interval-noncrossing if it is noncrossing and all -legs are outer; I NC denotes the set of all interval-noncrossing partitions, • noncrossing-interval if it is interval-noncrossing after swapping the colors and ; NC I denotes the set of all noncrossing-interval partitions, • interval-arbitrary if all -legs are outer; I A denotes the set of all interval-arbitrary partitions, • arbitrary-interval if it is interval-arbitrary after swapping the colors and ; A I denotes the set of all arbitrary-interval partitions, Hasse diagram of all two-colored admissible sets of partitions.
• noncrossing-arbitrary if every block that contains an inner -leg is monochrome and does not cross any other block, i.e. for all legs i, j, k, ℓ and all blocks NC A denotes the set of all noncrossing-arbitrary partitions, • arbitrary-noncrossing if it is noncrossing-arbitrary after swapping the colors and ; A NC denotes the set of all arbitrary-interval partitions, • pure noncrossing if it is noncrossing and all inner blocks are monochrome; pNC denotes the set of all pure noncrossing partitions, • pure crossing if connected inner legs have the same color; pC denotes the set of all pure noncrossing partitions, • arbitrary without any conditions; the set of all bipartitions is also denoted A .Theorem 6.10.There are exactly 12 admissible sets of 2-faced partitions (9 if we identify a set with the one obtained by simply swapping the two colors), namely those given in Definition 6.9. Figure 1 displays their respective containment by means of a Hasse diagram and gives minimal generating sets of 2-block partitions.
Proof.We know that a set obtained from a positive symmetric 2-faced universal product is automatically admissible.Of course, swapping the two colors turns an admissible set into an admissible set.This helps to settle admissibility of a large number of sets in the diagram: • The sets I , NC , A are the sets of interval, noncrossing, and all partitions (ignoring the colors), and thus are known to come from the trivially two-faced Boolean, free and tensor universal product, respectively.Swapping the colors does not change these sets of partitions.• The set NC I is the set of noncrossing-interval partitions, which originates from free-Boolean independence [Liu19].Swapping the colors leads to the set I NC .• The set A I comes from tensor-Boolean independence [GHU23].Swapping the colors leads to the set I A .• The set biNC is the set of binoncrossing partitions, it comes from bifree independence [CNS15,Voi14].Swapping the colors does not change the set.
We are left with the sets of pure crossing and pure noncrossing partitions and with the sets of noncrossing-arbitrary and arbitrary-noncrossing partitions, where again by swapping the colors it is enough to deal with the noncrossing-arbitrary ones.All properties are easily verified.
The theorem now follows from the fact that each admissible set is uniquely determined by which basic two-block partitions have nonzero coefficients, and from the implications in Corollary 6.8.□ Corollary 6.11.Let ⊙ be a positive symmetric 2-faced universal product.Then the admissible set of partitions Π ⊙ := {π ∈ P : α π ̸ = 0} is one of the 12 given in Definition 6.9.Furthermore: • If Π ⊙ ∈ {NC A , A NC , pNC , pC }, then the highest coefficients of ⊙ are given by the indicator function of Π ⊙ , and ⊙ does not coincide with any of the positive symmetric two-faced universal product given in [GHU23, Propositions 5.13 and 6.19].• In all other cases, ⊙ does coincide with one of the positive symmetric two-faced universal product given in [GHU23, Propositions 5.13 and 6.19]; more concretely, Proof.If ⊙ is a positive symmetric universal product, then its highest coefficients form an admissible family of weights.If all the basic coefficients are 0 or 1, the family must be given by the indicator function of one of the admissible sets of partitions and all except the mentioned four are identified as positive products in [GHU23]: • A corresponds to the tensor product • NC corresponds to the free product • biNC corresponds to the bifree product • I A and A I corresponds to the Boolean-tensor and tensor-Boolean product, respectively • I NC and NC I corresponds to the Boolean-free and free-Boolean product, respectively • I corresponds to the Boolean product If one of the basic coefficients is not 0 or 1, Lemma 6.7 leaves only three possibilities, in each of which the universal product has been found to be positive in [GHU23]: • ν = ξ = q ∈ T\{1}, in this case all other basic coefficients are forced to be equal to 1; by comparison of the basic coefficients, the corresponding universal product is the deformed tensor product with ζ = q, • ν = q ∈ T \ {1}, ξ = 0; in this case, the product must coincide with the deformed free product with ζ = q, • ν = 0, ξ = q ∈ T \ {1}; in this case, the product must coincide with the deformed bifree product with ζ = q.□ Remark 6.12.A remarkable property of freeness is that the free product of traces is again a trace.We cannot expect such a behaviour for any non-trivial multi-faced independence.Indeed, this would force the highest coefficients to be invariant under cyclic permutations, and since we may change the color of the first leg, we could change the color of every leg without changing the coefficient.
Remark 6.13.Bifreeness allows to define a convolution for probability measures on R 2 .This comes from the fact that for bifree pairs (a 1 , a 2 ), (b 1 , b 2 ) one always has commutativity of a 1 with b 2 and of a 2 with b 1 .Consequently, a 1 + b 1 commutes with a 2 + b 2 whenever a 1 , a 2 commute and b 1 , b 2 commute.If independent variables in different faces commute, one must have ξ = 1, which is only the case for tensor and bifree independence.
Remark 6.14.There are other interesting symmetric two-faced universal products which are not positive, for example the bi-Boolean product.It seems very well possible to do a classification under slightly relaxed conditions, only assuming that one is allowed to change the color of the first leg and not assuming any mirror symmetry (recall that we used changing the color on both sides to show that highest coefficients for all interval partitions are 1).However, it is not clear how to motivate those properties when one does not aim for positivity.For the construction of a universal product in the algebraic sense (see Section 8), Conditions (v) and (vi) of Theorem 5.3 are not necessary at all.

Moment-cumulant relations
A key tool in the proofs of the subsequent sections are cumulants.In this section, we adapt the theory of cumulants developed in [MS17] to our special case of symmetric multi-faced independences.
Observation 7.1.Let α be a family of weights such that α π is invertible for every one-block partition.For every family of moments, indeed, existence and uniqueness of the c f follows by a standard induction argument.Obviously, the cumulants also determine the moments.
If α is invariant under permutation of blocks, then the formula simplifies to There is no problem extending formulas (4) and (5) to a multivariate situation.To this end, we think of the (multivariate) moments and cumulants as linear functionals m, c : A → C, where A = C⟨x i : ∈ F, i ∈ I ⟩ is a multi-faced polynomial algebra with (possibly) several indeterminates x i , i ∈ I , for each face ∈ F. For a monomial X = x i(ℓn) .Cumulants are then defined by the relations respectively.In case each I is a one-element set, writing x for the indeterminates, formulas (6) and ( 7) are recovered by setting Definition 7.2.An algebraic probability space is a pair (A, Φ), where A is an algebra and Φ : A → C is a linear functional.Definition 7.3.Let (A, Φ) be an algebraic probability space and α = (α π ) π∈P (<) a family of weights on (ordered) multi-faced partitions.For a family a = (a i : ∈ F, i ∈ I ) ⊂ A, put j a : C⟨x i : ∈ F, i ∈ I ⟩ → A, x i → a i .We define its moments by m a (X) := Φ(j a (X)) and its α-cumulants c a (X) according to the moment-cumulant relations (6) or (7), respectively.Definition 7.4.Fix monic weights (α π ) π∈P .Let V = ∈F V be a vector space with a direct sum decomposition into subspaces according to the faces.Recall that T 0 (V ) = n∈N V ⊗n = ∈F T 0 (V ) denotes the (non-unital) free algebra over V and T (V ) := n∈N0 V ⊗n = C1 ⊕ T 0 (V ) its unitization, the free unital algebra over V .On the dual space T 0 (V ) ′ = {φ : T 0 (V ) → C linear} we define for where for π = {β 1 , . . ., β n } we put x Then exp α is a bijection.We denote the inverse simply as log α , which can be calculated recursively, Note that often exp α and log α are interpreted as bijections between linear functionals on T (V ) vanishing on 1 and unital linear functionals on T (V ) by extending the linear functionals from T 0 (V ) to T (V ) accordingly (i.e.ψ and log α (φ) are extended by annihilating the unit, while exp α (ψ) and φ are extended as unital maps).
We use the following conventions.
• If the weights α come from a universal product ⊙, we write exp ⊙ := exp α and log ⊙ := log α .
• If (x i ) i∈I form a basis of V , we identify T (V ) and T 0 (V ) with the noncommutative (unital or non-unital) polynomial algebras C⟨x i : i ∈ I , ∈ F⟩ and C⟨x i : i ∈ I , ∈ F⟩ 0 , respectively.
Definition 7.5.Let A be a multi-faced algebra and φ : A → C a linear functional.We define Â := T 0 ∈F A and φ := φ • µ, where µ : T 0 ∈F A → A is the canonical homomorphism.
Observation 7.6.Let α be monic weights.Let furthermore φ : A → C be a linear functional on a multi-faced algebra A and a = (a i ∈ A : i ∈ I , ∈ F) a family of elements.With the notations from the previous definitions, for ). Observation 7.7.Let h : B → A be an F-faced homomorphism between F-faced algebras B, A and define ĥ : B → Â as the unique algebra homomorphism with ĥ(b) = h(b) for all b ∈ B , ∈ F. Automatically, ĥ is an F-faced homomorphism and fulfills Therefore, given monic weights (α π ) π∈P , one finds that Theorem 7.8 (Adjusted and simplified from [MS17, Th. 7.2]).A positive and symmetric universal product is uniquely determined by its highest coefficients.More precisely, for a here we use the direct sum as a shorthand notation for the corresponding linear functional on T 0 κ∈[2], ∈F A i = Â1 ⊔ Â2 as described by Equation (1).
Proof.We only explain why this is a special case of [MS17, Th. 7.2] and refer the reader to [Var21, Theorems 2.4.12 and 2.5.13] for a detailed discussion.Since ⊙ is positive, their are no wrongordered highest coefficients.In the symmetric case, the exponential and logarithm map used in [MS17] coincide with the maps of Definition 7.4 and are therefore determined by the highest coefficients.Since ⊙ is symmetric, the second ingredient which is in general needed to determine the universal product, namely the nth order cumulant Lie algebra, is trivial for all n.□

Reconstruction of universal products from highest coefficients
In this section we prove that every admissible family leads to a unique universal product.In particular, we can associate universal products with the admissible sets NC A , A NC , pNC , pC .However, it remains an open problem at the moment to decide whether or not those universal products are positive.
We define a modified family ã = (a ℓ : ∈ F, ℓ ∈ Ĩ ) where Ĩ := I \ {i, i + 1} ∪ {{i, i + 1}} and a {i,i+1} := a i a i+1 .For X := x f (1) 1 the moments and cumulants according to Definition 7.3 fulfill m ã( X) = m a (X) and Proof.The claimed equality for the moments is obvious.The claim for the cumulants is proved by induction on n.For n = 2, i.e.X = x 1 x 2 , we have For general n, we can use the moment-cumulant relations for m a (X) = m ã( X) and obtain where we used α ρ = α π α {β1,β2} for π = ρ β1⌣β2 .On the other hand, with f ∈ F Recall that there is a canonical bijection between partitions σ ∈ P( f ) and partitions π ∈ P(f ) with i, i + 1 in the same block β ∈ π.Also, the highest coefficients α σ and α π agree under this bijection by Theorem 5.3 (iii).Using the induction hypothesis on c a (X ↾ β) finishes the proof.□ Theorem 8.2.Suppose that the weights (α π ) π∈P are admissible.Then there exists a unique symmetric universal product with highest coefficients (α π ) π∈P .
Proof.The uniqueness statement is proved in [MS17, Th. 7.2], see Theorem 7.8.Let φ κ : A κ → C be linear functionals on 2-faced algebras (κ ∈ {1, 2}).Recall Definition 7.4 of exp α and log α and Definition 7.5, which sets the notation for lifting φ k to linear functionals Let us say that a partition π is adapted to b, and write π ≺ b, if b is constant on blocks of π (this is the first condition of π being adapted to s).Note that (log φ1 ⊕ log φ2 ) ⊗|π| (a π ) = 0 when π is not adapted to b; indeed, this follows directly from the way we identify the direct sum of linear functionals with a linear functional on the tensor algebra in Equation (1).With this in hand, we calculate On the other hand, if the two legs i and i + 1 are not identified, then there are partitions adapted to b for which i, i + 1 lie in the same block as well as ones for which i, i + 1 lie in different blocks.This leads to using α π α {β1,β2} = α σ when σ = π \ { β} ∪ {β 1 , β 2 }, i.e. π = σ β1⌣β2 .The two expressions derived in (9) and (10) agree by Lemma 8.1 and, therefore, we have a well-defined map Let us verify that ⊙ is indeed a symmetric universal product.To prove universality, recall Observation 7.7.Let h κ : B κ → A κ be F-faced algebra homomorphisms and φ κ : Symmetry and unitality are immediate for ⊙ and therefore descend to ⊙.To prove associativity is slightly more involved.We write and claim that the unique algebra homomorphism extending the canonical embeddings therefore log φ 1 ⊙ φ 2 = (log φ1 ⊕ log φ2 ) • λ 12 as claimed.The rest is easy: note that λ 12 is actually a projection onto a subalgebra, so we can safely identify b with the corresponding element in the domain of λ 12 ⊔ id instead of introducing yet another symbol for its preimage.The other direction, i.e.
follows by symmetry.
To check that the highest coefficients of ⊙ are indeed given by α, it is enough to consider products of two functionals φ as needed. 11□ The formula to compute mixed moments can be considerably simplified in the special case where the the highest coefficients are only 0 or 1.
does not influence the result!Definition 8.3.We say that a symmetric universal product is combinatorial with partition set Π if its highest coefficients are all either 0 or 1 and Π = {π ∈ P : α π = 1}.
Theorem 8.4.Let ⊙ be a combinatorial universal product with admissible partition set Π one of the 12 sets of Theorem 6.10 (in particular, F = { , } a two element set).Furthermore, let φ κ be a linear functional on a multi-faced algebra • π ∈ P(f ) the multi-faced partition with blocks β κ := {i : b(i) = κ} (whenever non-empty), • Π ≤π := {σ ∈ Π : σ ≤ π} the set of refinements of π inside Π ∩ P(f ), • S the set of maximal elements of Π ≤π (i.e.coarsest refinements of π inside Π), • ∧R is the maximal common refinement of partitions in R ⊂ Π ∩ P(f ), ∧∅ := 1 f , The key observation is that a refinement σ of a partition ρ ∈ Π belongs to Π if and only if σ ↾ β ∈ Π for all blocks β ∈ ρ; this can be easily seen for each of the 12 admissible sets of partitions individually.Using the moment cumulant formula on each block of ∧R and the observation on refinements just made, we find Recall that we defined ∧∅ := 1 f , so that this confirms that the choice is consistent with Equation (11), and it also shows that the statement of the theorem is equivalent to LHS of (11) = 0. □ Example 8.5.Let ⊙ be the universal product associated with NC A .Then has set of coarsest refinements

Unit preserving universal products
In [DAGSV22], Diaz-Aguilera, Gaxiola, Santos, and Vargas characterize when the moment cumulant relation associated with weights on partitions leads to independent constants, finding this to be the case if and only if the weights do not change when removing or inserting a singleton from or to the partition.Manzel and Schürmann discuss in [MS17,Rem. 3.1] the relation between universal products in the category of multi-faced algebras and in the category of multi-faced unital algebras and observe that while a product for the unital category always gives rise to a product for the non-unital category, the other way round requires a condition, namely that the universal product respects the units or is unit preserving as we prefer to write in this article.In this section we briefly review universal products in the category of multi-faced unital algebras, define what exactly it means to be unit preserving, generalize the definition of singleton inductive weights to the multi-faced setting, and finally characterize unit preserving symmetric universal product as those whose highest coefficients are singleton inductive.
In the category of unital algebras with unital algebra homomorphisms, the coproduct is given by the unital free product, which can be constructed from the non-unital free product as here ⟨•⟩ denotes the generated two-sided ideal.
• A multi-faced unital algebra is a unital algebra A with unital subalgebras A , ∈ F, such that the canonical unital algebra homomorphism 1 ∈F A → A is an isomorphism, in which case we write A = 1 ∈F A .• A multi-faced unital algebra homomorphism is a unital algebra homomorphism which maps face into face.• The unital free product of multi-faced unital algebras A 1 , A 2 is a multi-faced unital algebra with (A 1 ⊔ 1 A 2 ) := A 1 ⊔ 1 A 2 .• A linear functional ϕ : A → C on a multi-faced unital algebra is unital if ϕ(1 A ) = 1.
Multi-faced unital algebras with multi-faced unital algebra homomorphisms form a category, in which ⊔ 1 is a coproduct.One can adapt Definition 3.1 to the unital situation and obtains the following.
Definition 9.2.A universal product in the category of multi-faced unital algebras is a binary product operation for unital linear functionals on multi-faced unital algebras which associates with unital functionals ϕ 1 , ϕ 2 on multi-faced unital algebras A 1 , A 2 , respectively, a unital functional for all multi-faced unital algebra homomorphisms As Manzel and Schürmann noticed in [MS17, Rem.3.1], every universal product ⊙ in the category of multi-faced unital algebras gives rise to a universal product in the sense of Definition 3.1, simply putting φ 1 ⊙ φ 2 := φ1 ⊙ φ2 ↾ A 1 ⊔ A 2 ⊂ Ã1 ⊔ 1 Ã2 where Ãi denotes the unitization of a multi-faced algebra and φi the unital extension of a linear functional.
Definition 9.3.A universal product is unit preserving (or respects units) if, whenever A 1 , A 2 are multi-faced algebras with each A i unital and φ i a linear functional on A i which vanishes on the ideal ⟨1 i − 1 i : , ∈ F⟩ ⊂ A i and such that φ i ↾ A i is unital for every ∈ F, then φ 1 ⊙ φ 2 vanishes on the ideal ⟨1 i − 1 j : i, j ∈ [2], , ∈ F⟩ ⊂ A 1 ⊔ A 2 and φ 1 ⊙ φ 2 ↾ A i is unital for every i ∈ [2], ∈ F. Assume that ⊙ is unit preserving.The φ i = ϕ i • p Ai in (12) are linear functionals on A i , vanish on ker p Ai = I Ai = ⟨1 i − 1 i : , ∈ F⟩ ⊂ A i and fulfill φ i (1 i ) = ϕ i (1 i ) = 1.Therefore, we may conclude that φ 1 ⊙ φ 2 vanishes on the ideal ⟨1 i − 1 j : i, j ∈ [2], , ∈ F⟩ ⊂ A 1 ⊔ A 2 , which coincides with the kernel of the canonical homomorphism p : A 1 ⊔ A 2 → A 1 ⊔ 1 A 2 .This means that there is a well-defined linear functional ϕ 1 ⊙ ϕ 2 with φ 1 ⊙ φ 2 = (ϕ 1 ⊙ ϕ 2 ) • p.This functional is also unital because We leave the rest of the simple, but notationally cumbersome proof of the claim (in particular universality and associativity of ⊙) to the interested reader.
In the following we will need often remove a singleton block β = {s} from a partition π ∋ β.While consistent use of notation would dictate to write π \ {β} = π \ {{s}}, we will prefer to write π \ {s} for better legibility.and analogously log α φ(a ⊗ 1 ) = 0. Now assume the statement holds for all 1 < m < n and consider a = a 1 ⊗ • • • ⊗ a n with a i ∈ A f (i) , a s = 1 f (s) .Note that φ(a (here ǎs means omission of the factor) and log α φ(a s ) = log α φ(1 ) = φ(1 ) = 1.We find Theorem 9.7.For a multi-faced positive symmetric universal product ⊙, the following are equivalent.
A simple induction on the number of blocks shows that α π = ν • α π\{s} whenever π ∈ P has a singleton block {s} ∈ π of color .Therefore, ν = 1 for all ∈ F implies that the highest coefficients are singleton inductive.Now assume that the highest coefficients of a positive symmetric universal product are singleton inductive.Let A 1 , A 2 be multi-faced algebras with unital faces, s = b × f ∈ ([2] × F) n , a ℓ ∈ A
Lemma 8.1.Suppose that the weights (α π ) π∈P are admissible.Fix a family of elements a = (a ℓ : ∈ F, ℓ ∈ I ) ⊂ A in an algebraic probability space (A, Φ) such that [n] is the disjoint union of the I .Put f (ℓ) := if ℓ ∈ I and assume that f

Remark 9. 4 .
A multi-faced universal product is unit preserving if and only if (12) is well-defined, in which case it yields a universal product in the category of multi-faced unital algebras [MS17, Rem.3.1].Since Manzel and Schürmann do not give a definition of "respecting units", let us briefly check that Definition 9.3 captures what they mean.